Dynamic programming for graphs on surfaces
We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k) · n steps. Our approach combines tool...
| Autores: | , , |
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| Tipo de recurso: | artículo |
| Fecha de publicación: | 2014 |
| País: | España |
| Institución: | Universitat Politècnica de Catalunya (UPC) |
| Repositorio: | UPCommons. Portal del coneixement obert de la UPC |
| Idioma: | inglés |
| OAI Identifier: | oai:upcommons.upc.edu:2117/86631 |
| Acceso en línea: | https://hdl.handle.net/2117/86631 https://dx.doi.org/10.1145/2556952 |
| Access Level: | acceso abierto |
| Palabra clave: | Graph Algorithms analysis of algorithms parameterized algorithms graphs on surfaces branchwidth dynamic programming polyhedral embeddings noncrossing partitions Grafs, Teoria de Àrees temàtiques de la UPC::Matemàtiques i estadística |
| Sumario: | We provide a framework for the design and analysis of dynamic programming algorithms for surface-embedded graphs on n vertices and branchwidth at most k. Our technique applies to general families of problems where standard dynamic programming runs in 2O(k·log k) · n steps. Our approach combines tools from topological graph theory and analytic combinatorics. In particular, we introduce a new type of branch decomposition called surface cut decomposition, generalizing sphere cut decompositions of planar graphs, which has nice combinatorial properties. Namely, the number of partial solutions that can be arranged on a surface cut decomposition can be upper-bounded by the number of noncrossing partitions on surfaces with boundary. It follows that partial solutions can be represented by a single-exponential (in the branchwidth k) number of configurations. This proves that, when applied on surface cut decompositions, dynamic programming runs in 2O(k) · n steps. That way, we considerably extend the class of problems that can be solved in running times with a single-exponential dependence on branchwidth and unify/improve most previous results in this direction. |
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